- #1

bobinthebox

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- TL;DR Summary
- How is the divergence defined for such a tensor field?

Hi everyone,

studying the bending of an incompressible elastic block of Neo-Hookean material, one finds out the first Piola-Kirchoff stress tensor as at page 182 (equation 5.93)

where $e_r = cos(\theta)e_1 + \sin(\theta)e_2$ and $e_{\theta} = -sin(\theta)e_1 + \cos(\theta)e_2$

How is the divergence defined in this case? The book writes the following

$$Div(S) = \frac{\partial }{\partial x_1^0} S_{r1} e_r + \frac{\partial}{\partial x_2^0} S_{\theta 2} e_{\theta} = 0$$

but I can't see why this is the divergence of $S$. I'd like to have a formal proof of this. I know that, by definition, the divergence of a tensor field $S$ is the unique vector field such that

$Div(S^T a)=Div(S) \cdot a$

for every vector $a$

studying the bending of an incompressible elastic block of Neo-Hookean material, one finds out the first Piola-Kirchoff stress tensor as at page 182 (equation 5.93)

where $e_r = cos(\theta)e_1 + \sin(\theta)e_2$ and $e_{\theta} = -sin(\theta)e_1 + \cos(\theta)e_2$

How is the divergence defined in this case? The book writes the following

$$Div(S) = \frac{\partial }{\partial x_1^0} S_{r1} e_r + \frac{\partial}{\partial x_2^0} S_{\theta 2} e_{\theta} = 0$$

but I can't see why this is the divergence of $S$. I'd like to have a formal proof of this. I know that, by definition, the divergence of a tensor field $S$ is the unique vector field such that

$Div(S^T a)=Div(S) \cdot a$

for every vector $a$